On families of holomorphic differentials on degenerating annuli

TL;DR

This paper investigates the local analytic behavior of holomorphic differentials on degenerating annuli, providing key lemmas and isomorphisms that describe their extension and structure in degenerating families of Riemann surfaces.

Contribution

It introduces a normal families lemma, an isomorphism of sheaves for regular differentials, and a holomorphic extension property for degenerating families, advancing understanding of differentials on degenerating curves.

Findings

Normal families lemma established

Isomorphism of sheaves for k-differentials derived

Extension property for families to nodal curves proven

Abstract

We consider the local analytic behavior for a family of holomorphic differentials on a family of degenerating annuli. Three results and discussion are presented. The first is the normal families Lemma 1. The second is an isomorphism of sheaves, formula (3), giving a direct description of families of regular $k$-differentials (sections of powers of the relative dualizing sheaf) in terms of $k$-canonical forms on the total space of the family. The third is a general holomorphic extension property, Lemma 3, for families given on smooth Riemann surfaces/curves to extend to the limiting nodal Riemann surfaces/curves. Divisors of families of holomorphic differentials are also discussed.